( which did not exist in Diophantus's time ), his method would be visualised as drawing a tangent to a curve at a known rational point, and then finding the other point of intersection of the tangent with the curve ; that other point is a new rational point.

Analytically, x can also be raised to an irrational power ( for positive values of x ); the analytic properties are analogous to when x is raised to rational powers, but the resulting curve is no longer algebraic, and cannot be analyzed via algebraic geometry.

These solutions yield good rational approximations of the form x / y to the square root of n. In Cartesian coordinates, the equation has the form of a hyperbola ; it can be seen that solutions occur where the curve has integral ( x, y ) coordinates.

It is clear that a 0 % tax rate raises no revenue, but the Laffer curve hypothesis is that a 100 % tax rate will also generate no revenue because at such a rate there is no longer any incentive for a rational taxpayer to earn any income, thus the revenue raised will be 100 % of nothing.

The definition of elliptic curve from algebraic geometry is connected non-singular projective curve of genus 1 with a given rational point on it.

In 1976, M. Rosen showed how to realize any countable abelian group as the class group of a Dedekind domain which is a subring of the rational function field of an elliptic curve, and conjectured that such an " elliptic " construction should be possible for a general abelian group ( Rosen 1976 ).

* Normal degree of a rational curve on a surface.

In number theory, the Mordell conjecture is the conjecture made by that a curve of genus greater than 1 over the field Q of rational numbers has only finitely many rational points.

* Case g = 1: no points, or C is an elliptic curve and its rational points form a finitely generated abelian group ( Mordell's Theorem, later generalized to the Mordell – Weil theorem ).

* The Mordell conjecture that a curve of genus greater than 1 over a number field has only finitely many rational points ;

The theorem states that any elliptic curve over Q can be obtained via a rational map with integer coefficients from the classical modular curve

A well-known example is the Taniyama – Shimura conjecture, now the modularity theorem, which proposed that each elliptic curve over the rational numbers can be translated into a modular form ( in such a way as to preserve the associated L-function ).

Rational Bézier curve – polynomial curve defined in homogeneous coordinates ( blue ) and its projection on plane – rational curve ( red )

It applies to an elliptic curve E, and the problem it attempts to solve is the prediction of the rank of the elliptic curve over the rational numbers ( or another global field ): i. e. the number of free generators of its group of rational points.

The evaluation map sends the fundamental class of M to a d-dimensional rational homology class in Y, denoted

If the transfer function is a rational function with real poles and zeros, then the Bode plot can be approximated with straight lines.

The existence of irreducible polynomials of degree greater than one ( without zeros in the original field ) historically motivated the extension of that original number field so that even these polynomials can be reduced into linear factors: from rational numbers ( ), to the real subset of the algebraic numbers ( ), and finally to the algebraic subset of the complex numbers ( ).

Additionally, consider for instance the unit circle S, and the action on S by a group G consisting of all rational rotations.

* G. W. F. Hegel: Emphasized the " cunning " of history, arguing that it followed a rational trajectory, even while embodying seemingly irrational forces ; influenced Marx, Kierkegaard, Nietzsche, and Oakeshott.

Some philosophers believe that the " no-free-lunch in search and optimization theorem " of David Wolpert and William G. Macready is a probability-based extension of induction, yet this is misleading, as inductive logic accustomed to probabilistic arguments and the No free lunch theorem ( NFL ) is more a variation of economic rational choice theory.

* The irrational numbers are a G < sub > δ </ sub > set in R, the real numbers, as they can be written as the intersection over all rational numbers q of the complement of

; Inverse problem of Galois theory: Given a group G, find an extension of the rational number or other field with G as Galois group.

Alternatively, the quantum group U < sub > q </ sub >( G ) can be regarded as an algebra over the field C ( q ), the field of all rational functions of an indeterminate q over C.

Similarly, the quantum group U < sub > q </ sub >( G ) can be regarded as an algebra over the field Q ( q ), the field of all rational functions of an indeterminate q over Q ( see below in the section on quantum groups at q

Another example: P being as above, a resolvent R for a group G is a polynomial whose coefficients are polynomials in the coefficients of p, which provides some information on the Galois group of P. More precisely, if R is separable and has a rational root then the Galois group of P is contained in G. For example, if D is the discriminant of P then is a resolvent for the alternating group.

That implies that any two rational functions F and G, in the function field of the modular curve, will satisfy a modular equation P ( F, G ) = 0 with P a non-zero polynomial of two variables over the complex numbers.

Consider the unit circle S, and the action on S by a group G consisting of all rational rotations.

* For any non-zero invariant vector in a rational representation on G, there is an invariant homogeneous polynomial that does not vanish on it.

Hilbert had shown that this question is related to a rationality question for G: if K is any extension of Q, on which G acts as an automorphism group and the invariant field K < sup > G </ sup > is rational over Q, then G is realizable over Q.

Some of this is based on constructing G geometrically as a Galois covering of the projective line: in algebraic terms, starting with an extension of the field Q ( t ) of rational functions in an indeterminate t. After that, one applies Hilbert's irreducibility theorem to specialise t, in such a way as to preserve the Galois group.

Hegelianism is a collective term for schools of thought following or referring to G. W. F. Hegel's philosophy which can be summed up by the dictum that " the rational alone is real ", which means that all reality is capable of being expressed in rational categories.

Berlin contended that under the influence of Plato, Aristotle, Jean-Jacques Rousseau, Immanuel Kant and G. W. F. Hegel, modern political thinkers often conflated positive liberty with rational action, based upon a rational knowledge to which, it is argued, only a certain elite or social group has access.

Applying a result of MacIntyre on the model theory of p-adic integers, one deduces again that ζ < sub > G </ sub >( s ) is a rational function in p < sup >− s </ sup >.

Harold Pinter writes that Raymond G. H. Seitz: " had a very good reputation as a rational, responsible and highly sophisticated man.

Since C is rational, this correspondence has K coincidences, each of which implies a line of the pencil which meets its image.

This involves a sifting of the empirical and rational elements entering into each social science statement.

Kant argued for the establishment of a peaceful world community, not in a sense of a global government, but in the hope that each state would declare itself a free state that respects its citizens and welcomes foreign visitors as fellow rational beings, thus promoting peaceful society worldwide.

According to Weber, Confucianism and Puritanism are mutually exclusive types of rational thought, each attempting to prescribe a way of life based on religious dogma.

then each rational solution x,

As a consequence of the Weierstrass approximation theorem, one can show that the space C is separable: the polynomial functions are dense, and each polynomial function can be uniformly approximated by one with rational coefficients ; there are only countably many polynomials with rational coefficients.

Since they are produced automatically without any rational analysis and verification ( see the modern idea of the subconscious ) of whether they are correct or not, they need to be confirmed ( epimarteresis: confirmation ), a process which must follow each assumption.

Therefore, the rational decision for each voter is to be generally ignorant of politics and perhaps even abstain from voting.

** An individual voter may have a rational ignorance regarding politics, especially in nationwide elections, since each vote has little weight.

The rational powers and abilities of each and every human being were attributed to his soul, which was a genius.

Field Marshal Viscount Alanbrooke, Chief of the Imperial General Staff and co-chairman of the Anglo-US Combined Chiefs of Staff Committee for most of the Second World War, described the art of military strategy as: " to derive from the aim a series of military objectives to be achieved: to assess these objectives as to the military requirements they create, and the pre-conditions which the achievement of each is likely to necessitate: to measure available and potential resources against the requirements and to chart from this process a coherent pattern of priorities and a rational course of action.

“ to derive from the aim a series of military objectives to be achieved: to assess these objectives as to the military requirements they create, and the pre-conditions which the achievement of each is likely to necessitate: to measure available and potential resources against the requirements and to chart from this process a coherent pattern of priorities and a rational course of action .”

EMH advocates reply that while individual market participants do not always act rationally ( or have complete information ), their aggregate decisions balance each other, resulting in a rational outcome ( optimists who buy stock and bid the price higher are countered by pessimists who sell their stock, which keeps the price in equilibrium ).

Hunter, unlike his contemporaries … sought the reason for each phenomenon ), but because it afforded him the opportunity, given his empirical rather than rational bent, to study his main interest-life, in all its forms.

Thus in the common context of polynomials with rational coefficients, a polynomial is irreducible if it cannot be expressed as the product of two or more such polynomials, each of them having a lower degree than the original one.

Scientists on the ground will use X-ray crystallography to study each protein's three-dimensional structure which, when determined, may aid in controlling each protein's activity through rational drug design.

Therefore the problem of computing derivatives, antiderivatives, integrals, power series expansions, Fourier series, residues, and linear functional transformations of rational functions can be reduced, via partial fraction decomposition, to making the computation on each single element used in the decomposition.

However, classical bargaining theory assumes that each participant in a bargaining process will choose between possible agreements, following the conduct predicted by the rational choice model.

* Pseudo algebraically closed field ( mathematics ), a field with geometric features, namely each variety over it has a rational point

The costs are allocated in a rational and systematic manner as depreciation expense to each period in which the asset is used, beginning when the asset is placed in service.

Lange has been called the poetical theologian par excellence: “ It has been said of him that his thoughts succeed each other in such rapid and agitated waves that all calm reflection and all rational distinction become, in a manner, drowned ” ( F. Lichtenberger ).

He defends the Thomistic position that human beings are essentially rational animals, each one miraculously created.